EN3: Introduction to Engineering

 

 

Teach Yourself Vectors

 

 

Division of Engineering

Brown University

 

 

3. How to calculate the magnitude of a vector in terms of its components

Let r be a vector and let

where (x,y,z) are three numbers specifying the Cartesian components of the vector r.  Find a formula for the length (or magnitude) of r in terms of (x,y,z).

 

Elementary geometry, my dear Watson. Consider the figure shown above.  Observe that the magnitude of r is equal to the distance from O to P.  Begin by calculating the distance from O to Q.  Observe that OQR is a right angled triangle, so Pythagoras’ theorem gives

.  Now observe further that OQP is a right angled triangle, so apply Pythagoras’ theorem again to see that

 

 

Problems

 

3.1 Calculate the magnitudes of each of the vectors shown below

(a)

(b)

(c)

 

3.2 For the truss shown below, find the magnitude of the position vector of C with respect to O.

 

3.3 A vector has magnitude 3, and i and j components of 1 and 2, respectively.  Calculate its k component.

 

3.4 Let {i,j,k} be a Cartesian basis.  A vector a has magnitude 4 and subtends angles of 30 degrees and 100 degrees to the i and k directions, respectively.  Calculate the components of a in the basis {i,j,k}